Long Cycles in Euclidean 2-Factors

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(suggested by William Cook)

Given a set S of points in the plane, a minimum 2-factor is a collection of circuits such that every point in S is a member of exactly one circuit in the collection and the sum of the Euclidean lengths of the edges in the circuits is as small as possible. An example of a minimum 2-factor is shown in the figure below, where the 10,000 points were selected uniformly at random in the unit square.

R10000.jpg

Notice that one of the circuits, colored red, contains a large fraction of the point set (2,264 of the 10,000 points). The problem is to show that this happens with high probability. That is, given a collection of n points in the unit square, selected uniformly at random, and a minimum 2-factor F, does F have with high probability a circuit of length at least cn for some positive constant c?